Question

Consider the given limits (a is a constant, f(x) ≥ 0). lim_(x->a) f(x) = 0 lim_(x->a) g(x) = 0 lim_(x->a) h(x) = 1 lim_(x->a) p(x) = infinity lim_(x->a) q(x) = infinity Evaluate each limit below. If a limit is indeterminate, enter INDETERMINATE. (If you need to use -[infinity] or [infinity], enter -INFINITY or INFINITY.)

Answer 1

We are required to evaluate the following.

$(a) lim_{x \rightarrow a} \dfrac{f(x)}{g(x)} \\(b) lim_{x \rightarrow a} \dfrac{f(x)}{p(x)} \\(c) lim_{x \rightarrow a} \dfrac{h(x)}{p(x)} \\(d)lim_{x \rightarrow a} \dfrac{p(x)}{q(x)}$

Answer:

(a)Indeterminate

(b)0

(c)0

(d)Indeterminate

Step-by-step explanation:

Given:

$lim_{x \rightarrow a} f(x) = 0\\ lim_{x \rightarrow a} g(x) = 0\\ lim_{x \rightarrow a} h(x) = 1\\ lim_{x \rightarrow a} p(x) = \infty\\ lim_{x \rightarrow a} q(x) = \infty$

Part A

$(a) lim_{x \rightarrow a} \dfrac{f(x)}{g(x)} =\dfrac{lim_{x \rightarrow a}f(x)}{lim_{x \rightarrow a}g(x)} \\=\dfrac{0}{0}=Indeterminate$

Part B

$lim_{x \right arrow a} \dfrac{f(x)}{p(x)} =\dfrac{lim_{x \rightarrow a}f(x)}{lim_{x \rightarrow a}p(x)} \\=\dfrac{0}{\infty}=0$

Part C

$lim_{x \rightarrow a} \dfrac{h(x)}{p(x)}

=\dfrac{lim_{x \rightarrow a}h(x)}{lim_{x \rightarrow a}p(x)} \\=\dfrac{1}{\infty}=0$

Part D

$lim_{x \rightarrow a} \dfrac{p(x)}{q(x)} =\dfrac{lim_{x \rightarrow a}p(x)}{lim_{x \rightarrow a}q(x)} \\=\dfrac{\infty}{\infty}=Indeterminate$

Answer 2

The limits which have indeterminate form , shown below;

                  $\lim_{x \to a} \frac{f(x)}{g(x)}=\frac{0}{0} \\\\\lim_{x \to a} \frac{p(x)}{q(x)}=\frac{\infty}{\infty}$

Indeterminate forms of limit:

An indeterminate form is an expression involving two functions whose limit cannot be determined solely from the limits of the individual functions.

Some Indeterminate form of limits are,

           0/0,  ∞/∞,  0·∞,  ∞−∞, ∞*0

It is given that,

               $\lim_{x \to a} f(x)=0\\\\ \lim_{x \to a} g(x)=0\\\\ \lim_{x \to a} h(x)=1\\\\ \lim_{x \to a} p(x)=\infty\\\\\lim_{x \to a} q(x)=\infty$

Now we have to find following;

      $\lim_{x \to a} \frac{f(x)}{g(x)}=\frac{0}{0} \\\\\lim_{x \to a} \frac{f(x)}{p(x)}=\frac{0}{\infty}\\\\\lim_{x \to a} \frac{h(x)}{p(x)}=\frac{1}{\infty}=0\\\\\lim_{x \to a} \frac{p(x)}{q(x)}=\frac{\infty}{\infty}$

Learn more about the limit of function here:

https://brainly.com/question/23935467